vol. 178, no. 5
the american naturalist
november 2011
Mussel Bed Boundaries as Dynamic Equilibria: Thresholds,
Phase Shifts, and Alternative States
Megan J. Donahue,1,2,* Robert A. Desharnais,1 Carlos D. Robles,1 and Patricia Arriola1,†
1. Department of Biological Sciences, California State University at Los Angeles, Los Angeles, California 90032; 2. Hawaii Institute of
Marine Biology, University of Hawaii, P.O. Box 1346, Kaneohe, Hawaii 96744
Submitted November 12, 2011; Accepted July 23, 2011; Electronically published October 5, 2011
Online enhancements: appendixes.
abstract: Ecological thresholds are manifested as a sudden shift
in state of community composition. Recent reviews emphasize the
distinction between thresholds due to phase shifts—a shift in the
location of an equilibrium—and those due to alternative states—a
switch between two equilibria. Here, we consider the boundary of
intertidal mussel beds as an ecological threshold and demonstrate
that both types of thresholds may exist simultaneously and in close
proximity on the landscape. The discrete lower boundary of intertidal
mussel beds was long considered a fixed spatial refuge from sea star
predators; that is, the upper limit of sea star predation, determined
by desiccation tolerance, fixed the lower boundary of the mussel bed.
However, recent field experiments have revealed the operation of
equilibrium processes that maintain the vertical position of these
boundaries. Here, we cast analytical and simulation models in a
landscape framework to show how the discrete lower boundary of
the mussel bed is a dynamic predator-prey equilibrium, how the
character of that boundary depends on its location in the landscape,
and how boundary formation is robust to the scale of local
interactions.
Keywords: threshold, boundary formation, stable states, hysteresis,
size-dependent predation.
Introduction
Ecological thresholds are manifested as a sudden shift in
state of community composition. Thresholds have received
increasing attention across many ecological systems (Wilson and Nisbet 1997; Robles and Desharnais 2002; Pascual
and Guichard 2005; Groffman et al. 2006; Mumby et al.
2009; Lauzon-Guay and Scheibling 2010; Osman et al.
2010) and may be characterized over space (e.g., a sudden
shift in vegetation pattern along a continuous rainfall gradient; van de Koppel and Rietkerk 2004) or time (e.g., a
* Corresponding author; e-mail: [email protected].
†
Present address: Department of Mathematics, Los Angeles City College, Los
Angeles, California 90029.
Am. Nat. 2011. Vol. 178, pp. 612–625. 䉷 2011 by The University of Chicago.
0003-0147/2011/17805-52611$15.00. All rights reserved.
DOI: 10.1086/662177
rapid shift from coral to algal dominance in response to
declining herbivore pressure; Mumby et al. 2009). Beginning with May (1977), theoretical approaches emphasized
thresholds that result from multiple stable equilibria. In
this case, two or more basins of attraction exist at a single
point in parameter space (Petraitis and Dudgeon 2004),
and initial conditions determine which equilibrium the
system achieves. Multiple (or “alternative”) stable states
are of particular interest and concern for ecological systems
exhibiting hysteresis: when a small parameter change results in a dramatic shift in system state that persists when
the parameter change is reversed. This characteristic of
ecological thresholds has garnered attention in conservation and ecosystem management because of the potential
for rapid, irreversible ecological collapse in response to
gradual environmental degradation (Muradian 2001;
Scheffer et al. 2001; Huggett 2005).
Recent reviews (Dudgeon et al. 2010; Petraitis and Hoffman 2010) have emphasized the need to distinguish ecological thresholds that result from a switch between alternative stable states and those that result from phase shifts.
Briefly (see Dudgeon et al. 2010 and Petraitis and Hoffman
2010 for a more detailed discussion), a phase shift is a
shift in the location of a single equilibrium in state space
in response to changes in environmental parameters; that
is, the underlying environment changes, and, therefore, so
does the community it can support. Phase shifts can be
dramatic if changes in the environmental parameters are
dramatic. In contrast, a switch between alternative stable
states is a shift between two distinct equilibria in state
space, both possible under the same environmental conditions, that can be difficult to reverse in the presence of
hysteresis. For ecological thresholds that arise from multiple stable states, the underlying mechanism typically depends on positive feedback interactions along an environmental gradient in time or space. Mumby et al. (2009)
models a coral-algal system where per capita mortality risk
from grazing decreases as macroalgal density increases
Mussel Bed Boundaries as Thresholds 613
(positive feedback) and grazing pressure declines in time
(temporal gradient in grazing). The model generates a
switch from multiple equilibria to a single, stable, macroalgal-dominant equilibrium as grazing pressure declines.
When multiple stable states give rise to thresholds in space
rather than in time, the spatial scale of positive feedback
must be local in order for densities to build up locally
(van de Koppel et al. 2005). For example, van de Koppel
and Rietkerk (2004) present a model of vegetation on an
arid landscape where rainwater capture is enhanced by
increased local vegetation density (positive feedback) on
a spatial gradient of rainfall, resulting in a sharp break in
the distribution of vegetation. Similarly, in a general model
presented by Wilson and Nisbet (1997), abrupt thresholds
in abundance arise when survivorship is enhanced by increased local density on a gradient of mortality.
Ecological thresholds arising from phase shifts and alternative stable states may also explain the characteristics
of species’ range limits (Holt et al. 2005). Typically, species’
range limits are gradual, reflecting a decline in density with
the continuous change of underlying niche axes (Holt et
al. 2005), that is, a spatial phase-shift threshold. However,
abrupt range limits can result from strong Allee effects: a
negative per capita growth rate at low density creates an
alternative stable equilibrium at zero density, which can
result in an abrupt species boundary on a continuous
gradient of habitat degradation (Holt et al. 2005), that is,
a spatial alternative-stable-state threshold. In this study,
we show (1) how continuous and abrupt species boundaries arise in the same underlying population model from
phase shifts and alternative equilibria, respectively, (2) how
these distinct threshold types can occur in close spatial
proximity on real landscapes, (3) how the existence and
location of the abrupt boundary depend on hysteresis, and
(4) how the character of the boundary depends on the
scale of local interactions.
Mussel beds have been a rich meeting ground for spatial
theory and empiricism (e.g., Paine and Levin 1981; Wootton 2001; Robles and Desharnais 2002; Guichard et al.
2003; van de Koppel et al. 2005). While prior landscape
models of mussel beds often focused on the dynamics and
consequences of gap formation (Wootton 2001; Guichard
et al. 2003), here we expand on the theory of boundary
formation in mussel beds (Robles and Desharnais 2002;
Donalson et al. 2004; Robles et al. 2010).
The original explanation of mussel bed limitation (Paine
1966, 1974; Connell 1972) maintained that the upper vertical boundary was set by physical stress (desiccation and
high temperature) while the lower boundary was set by
intense predation by the sea star Pisaster ochraceus. How
predation determined the lower boundary was of particular interest because the boundary remains stationary despite high interannual variation in mussel recruitment. The
shore level of the lower boundary was believed to be fixed
by the intolerance of Pisaster to exposure at low tide. Thus,
the upper shore provided a spatial refuge for the desiccation-resistant Mytilus californianus from desiccationsusceptible Pisaster. Recent work proposed (Robles and
Desharnais 2002) and experimentally demonstrated (Robles et al. 2009, 2010) that the lower boundary is a dynamic
equilibrium along continuous gradients of prey production and predator attack rate. The dynamic-equilibrium
concept is more consistent with observational and experimental data than is an inviolable refuge: Pisaster are often
observed foraging above the lower mussel bed boundary
(Robles et al. 1995), and recent experiments (Robles et al.
2009) demonstrate both the downward extension of the
lower boundary when Pisaster are removed (confirmation
of Paine’s 1974 result) and the upward retraction of the
lower boundary when Pisaster are added (contra the fixedrefuge hypothesis). It is this dynamic model of mussel bed
boundary formation that we investigate here.
To understand the dynamic properties of the MytilusPisaster model, we pair nonspatial ordinary differential
equations (ODE model) with a spatially explicit, stochastic
cellular automaton (CA model; Wilson and Nisbet 1997),
which simulates mussel bed structure on an idealized intertidal landscape. The ODE model is a mean-field approximation of the CA model and gives insight into the
equilibrium dynamics of the system for the full range of
parameter values. When run on a homogeneous landscape,
the CA model is a stochastic simulator of the ODE model,
but the flexibility of the CA model opens our investigation
to the effects of local interactions and heterogeneous landscapes on dynamics and pattern formation.
ODE Model
We consider here the dynamics of a single mussel bed and,
therefore, model Mytilus-Pisaster interactions as an open
system (fig. 1). At any location in the mussel bed, the
supply of Mytilus recruits is constant, and recruitment rate
depends on available space. A newly settled mussel grows
Figure 1: Schematic of model processes illustrating open recruitment, growth, and predation of mussels and immigration and emigration of sea stars.
614 The American Naturalist
Table 1: Parameter definitions and default values for mean-field model
Symbol (units)
Default value
s0 (mm)
j (cell⫺1 day⫺1)
k (cell⫺1)
b (day⫺1)
s⬁ (mm)
sm0 (day⫺1)
v (cells day⫺1)
c (cells mm⫺1)
I (cell⫺1 day⫺1)
e0 (mm day⫺2)
1.0
1.0 (.001–1.25)
1.0
.0007
200 (30–200)
.0001
1.0 (0–1)
.04
.01
5.0
Description
Size at recruitment
Recruitment rate
Saturation density of mussels
Decrease in mussel growth rate with size
Asymptotic size of a mussel
Background per capita mussel mortality rate
Per capita predator attack rate
Decrease in predator attack rate with mean mussel size
Per capita predator immigration rate
Per capita predator emigration constant per unit prey consumed
Note: For the three parameters (j, v, and s⬁) that vary in the two-dimensional gradient model, the maximum and minimum
values are listed in parentheses.
asymptotically to a maximum size and, in the process,
becomes increasingly resistant to predation (Paine 1976);
small mussels also become more resistant to predation
when surrounded by larger mussels (Bertness and Grosholtz 1985; Fong 2009). Sea stars immigrate into the system at a constant rate, but their emigration rate depends
on the availability of mussel prey, an expression of the
predator’s aggregative response to changing prey abundance (Robles et al. 1995). The ODE model assumes that
the environment is homogeneous, that is, that there are
no environmental gradients or local interactions affecting
recruitment, growth, or predation. In the ODE model, we
examine environmental heterogeneity by varying the
model parameters that determine settlement, growth, and
predator attack rate, which depend on the location in the
landscape. For example, at low tidal heights, mussels spend
more time submerged and therefore experience higher
growth rates and higher predation risk; in areas with direct
wave exposure, more water flow leads to higher recruitment, higher growth rates, and lower predation risk.
Our derivation of the mean-field ODE model (see app.
A, available online) begins with the McKendrick–von
Foerster equation for an age-structured population and
the von Bertalanffy function for individual growth. Following Nisbet et al. (1997), we introduce variables S(t) for
the mean size density of mussels (mean mussel size per
unit area, including empty cells of size 0) and N(t) for
mean numerical density of mussels (number of mussels
per unit area). Since the unit area is equal to one cell in
the CA model, N(t) is equivalent to the proportion of
space occupied by mussels, and S(t) can be thought of as
the average size of mussels when the average size includes
empty cells of size 0. Local spatial effects in the CA model
are replaced by these mean-field global values. The result
is a system of three ordinary differential equations (derived
in app. A) describing the dynamics of mean mussel size
density, mean mussel numerical density, and predator den-
sity in a well-mixed, spatially homogeneous patch (i.e.,
uniform tidal height and wave exposure):
dS(t)
p s 0 j(1 ⫺ N(t)k⫺1) ⫹ b(s ⬁ N(t) ⫺ S(t))
dt
(1a)
⫺ (m 0 ⫹ vP(t)e⫺cS(t))S(t),
dN(t)
p j(1 ⫺ N(t)k⫺1) ⫺ (m 0 ⫹ vP(t)e⫺cS(t))N(t),
dt
(1b)
dP(t)
p I ⫺ e 0 P(t)(vS(t)e⫺cS(t))⫺1.
dt
(1c)
Equations (1a) and (1b) describe, respectively, the change
in mean size density and mean numerical density through
time. Mean size density (eq. [1a]) changes through (1)
recruitment of size-s0 mussels into unoccupied space at
maximum recruitment rate j, which declines to 0 as the
mussel bed approaches its saturation density k p 1 mussel
cell⫺1; (2) von Bertalanffy growth of all mussels at maximum growth rate b, which declines as S(t) approaches its
maximum value s⬁N(t), the maximum individual size
times the proportion of occupied space; and (3) sizeindependent background mortality at rate m0 and sizedependent predator-induced mortality at maximum rate
v, which declines with increasing mean size density at an
exponential rate c. Mean mussel numerical density (eq.
[1b]) (1) increases with mussel recruitment at a maximum
recruitment rate j, which declines as the mussel bed approaches its saturation density k, and (2) decreases with
background mortality at rate m0 and with predator-induced
mortality at a maximum rate v, which declines exponentially with increasing mussel size density. Predators (eq.
[1c]) immigrate into the system at a constant rate I and
emigrate at a rate inversely proportional to the per capita
rate at which predators consume prey. Model parameters
are listed in table 1.
Mussel Bed Boundaries as Thresholds 615
Setting the time derivatives of equation (1) to 0 gives
the joint predator-prey equilibrium (S ∗, N ∗, P ∗). As shown
in appendix A, the equilibrium mean mussel size density
S ∗ is given by the roots of the following function:
f(S) p s 0 j ⫹
j(s ⬁ b ⫺ s 0 jk⫺1)
jk ⫹ m 0 ⫹ (Iv 2/e 0 )Se⫺2cS
⫺1
[
⫺ b ⫹ m0 ⫹
() ]
Iv 2 ⫺2cS
Se
S.
e0
Mussel Recruitment
(2)
Once the equilibrium value S ∗ has been obtained, the
remaining equilibrium values are given by
∗
P∗ p
N∗ p
( )
Ive⫺cS ∗
S ,
e0
change of the parameters over the gradients described below, specific functional forms are usually not known. In
the following description of parameterization, we chose
the simplest functional form indicated by existing data.
Default parameter values (table 1) were derived from the
literature and unpublished studies, as described below.
(3)
j
∗ .
jk ⫹ m 0 ⫹ (Iv/e 0 )S ∗e⫺2cS
⫺1
Spatial Model: Structure and Assumptions
Representation of Space
The spatially explicit cellular-automaton (CA) model is a
stochastic simulator of the ODE model to which environmental gradients and spatially local interactions can be
added. In the CA model, the intertidal landscape is represented as a rectangular lattice of cells, where each cell is
a potential site for a mussel. Each cell is identified by its
(x, y) coordinate in a w # h lattice, where the horizontal
dimension represents alongshore position and the vertical
dimension represents tidal height; x and y coordinates are
scaled to vary from 0 to 1. In this study, all simulations
are run on lattices of 2,000 # 2,000 cells. Each cell can
take on integer state values representing mussel size. The
state of a cell in position (x, y) at time t is Sxy 苸
{0, 1, … , s ⬁, xy}, where 0 is an empty cell, 1 is a cell with
a newly recruited mussel, and s⬁, xy is the asymptotic maximum size of a mussel at a particular (x, y) coordinate.
Model dynamics are described as stochastic transitions between states, where the transition probabilities for any
given cell are determined by the current state of the cell
(size effects), the states of the surrounding cells (neighborhood effects), and the cell’s position in the lattice (gradient effects). Gradient effects represent tidal exposure in
the y-dimension and wave energy in the x-dimension, such
that a cell’s position along these gradients determines
physical stress and the flux of recruits and food particles
to the rock surface. The transition probabilities therefore
represent position-specific mussel recruitment, growth to
larger sizes, and predator-induced mortality. While there
are numerous empirical studies indicating the direction of
Spatial patterns of mussel recruitment are, perhaps, the
least understood aspect of mussel ecology. Mytilus californianus recruitment probabilities often increase with
wave energy along a given shore level (Menge 1992; Robles
1997; Moya 2005). Plankton collections taken just offshore
(!20 m) from mussel beds revealed competent larvae concentrated in the top 2 m of the water column (C. D. Robles,
unpublished data). Given shore level variation in tidal
emersion and greater vertical mixing of wave-exposed
shores (Denny 1988; Denny and Shibata 1989), this observation predicts a vertical profile of settlement with a
midshore peak and a wider vertical spread on waveexposed shores. This prediction was confirmed by empirical measurements of mussel recruitment along tidalheight and wave-energy gradients that revealed a peak
occurring in the mid-intertidal zone at the wave-exposed
extreme and decreasing toward lower wave energies and
more extreme shore levels (Robles 1997; see app. B, available online). When the CA model includes spatial gradients, recruitment rates to empty cells peak at midshore
levels and high wave energies (fig. 2A). Along the vertical
axis, recruitment rate declines from the peak midshore
value to higher and lower shore levels according to a Gaussian-shaped curve. Along the horizontal axis, recruitment
rate declines exponentially with decreasing wave energy.
The gradient effects in recruitment are summarized by the
function
jxy p j1e⫺g(1⫺x)e⫺[(y⫺ym) /2n],
2
(4)
where g p ln (j1/j0 ), j0 and j1 are, respectively, minimum
and maximum recruitment rates along the wave-energy
gradient, ym is the midshore level where recruitment rates
reach a peak, and n is a parameter that specifies the rate
of recruitment decline with shore level moving away from
the midtidal peak.
Mussel Growth
Mussels have indeterminate, environmentally influenced
growth (Sebens 1987). Growth rates increase with longer
immersion times (i.e., lower shore levels; Dehnel 1956;
Garza 2005) and higher nutrient flux (i.e., higher wave
energies; Leigh et al. 1987; Dahlhoff and Menge 1996).
Growth proceeds asymptotically to maximum (“terminal”)
616 The American Naturalist
Mussel Mortality, Predation, and Neighborhood Effects
The CA model includes two possible sources of mussel
mortality: background mortality and predation. For simplicity, the same low rate of background mortality, m0, is
assumed for all mussels independent of their size and location in the lattice. Background mortality includes all
sources of mortality that are not explicitly from Pisaster
predation, including physical stress and other predators.
Experimental studies (Bertness and Grosholtz 1985; Fong
2009) indicate that younger, smaller mussels are shielded
from predators by the larger, less vulnerable members of
an aggregation. Thus, the CA model assumes that the likelihood of predation decreases as a mussel becomes surrounded by larger, less vulnerable individuals. Mortality
rates due to predation are modeled as the product of predator density, P(t), maximum per capita attack rate of predators at location (x, y), vxy, and the exponential decline in
predation due to a size-dependent neighborhood effect.
The total mortality rate of a mussel in cell (x, y) is given
by
mxy(t) p m 0 ⫹ vxy P(t)e⫺cSxy,r(t),
Figure 2: Landscape variation in recruitment rate j (A), maximum
size s⬁ (B), and predator attack rate v (C).
where c is the resistance to predation due to increasing
average mussel size in a neighborhood of radius r around
(x, y), Sxy, r(t). Neighborhood mean mussel size density
(including empty cells of size 0) is computed as a simple
average of the size of the mussels in the grid of (2r ⫹
1) # (2r ⫹ 1) cells centered at (x, y) in units of millimeters
per cell,
冘冘
x⫹r
sizes that vary along environmental gradients (Seed 1968,
1973; Kopp 1979; Robles et al. 2010). As in the ODE
model, mussel growth follows the von Bertalanffy function,
DSxy(t) p b(s ⬁, xy ⫺ Sxy(t)).
(5)
The expected growth increment of a mussel, DSxy, is proportional to the difference between the current size of a
mussel, Sxy, and its maximum asymptotic size, s⬁, xy, where
b is the maximum growth rate. As in the empirical studies,
the asymptotic size depends on flow rate and immersion
time and therefore increases with increasing wave energy
and decreasing shore level. For the spatial gradient in maximum size (fig. 2B), we use a simple product of linear
trends along the horizontal and vertical dimensions,
s ⬁, xy p s 1 ⫹ (s 2 ⫺ s 1)x(1 ⫺ y),
(6)
where s1 (30 mm) and s2 (200 mm) are the minimum and
maximum asymptotic sizes on the lattice and are based
on field measurements (Robles et al. 2010).
(7)
Sxy, r(t) p (2r ⫹ 1)⫺2
y⫹r
Sij (t).
(8)
ipx⫺r jpy⫺r
The maximum predation rate, vxy, varies with location
on the lattice. Empirically, predation rates decrease with
lengthening periods of tidal emersion (i.e., vertically toward higher shore levels; Paine 1974; Menge 1992; Robles
et al. 1995; Garza 2005) and greater hydrodynamic stress
(i.e., horizontally from sheltered to wave-exposed areas;
Menge 1983; Sanford 2002). To model the gradient in
maximum predator attack rate (fig. 2C), we use a simple
linear trend from a high predation rate v1, at the lowest
wave energy and lowest shore level, to a low rate v0, at the
highest wave energy and highest shore level:
vxy p v0 ⫹ (v1 ⫺ v0 )
2⫺x⫺y
.
2
(9)
Mussel mortality in the CA model is equivalent to that in
the ODE model when the neighborhood includes all cells
on the lattice (i.e., neighborhood mean mussel size density
equals global mean mussel size density) and when maxi-
Mussel Bed Boundaries as Thresholds 617
mum predation rate, vxy, is constant across the lattice (i.e.,
v0 p v1 in eq. [9]).
Predator Dynamics
Experimental evidence shows that sea stars aggregate when
there are episodes of massive recruitment of small mussels
and disperse when these preferred prey are reduced in
abundance relative to the larger mussels (Robles et al.
1995). These sea star aggregations arise because individuals
encountering masses of preferred (i.e., small) mussels suspend their usual vertical and alongshore foraging movements, thereby remaining in the concentration of preferred
prey, until these prey are depleted, at which point the
vertical and alongshore movements resume (Robles et al.
1995). Recent evidence indicates that Pisaster use tactile,
rather than waterborne, chemical cues for foraging (R.
Zimmer, personal communication), perhaps because waterborne cues are unlikely to give good directional information in high-water-motion environments. This makes
aggregative movement toward preferred prey less likely;
instead, we model predators with a constant immigration
rate and an emigration rate that depends on foraging
experience.
In our models, predator density is a global variable and
is expressed as an immigration-emigration process. We
assume that predators enter the system at a constant rate
I and exit the system at a per capita rate EP(t) that is
inversely proportional to the per capita rate of prey consumption,
E P(t) p
e0
(wh)⫺1 冘x, y Sxy(t)vxye⫺cSxy,r(t)
,
(10)
where the sum in the denominator is the total size density
of mussels lost to predation over all the cells in the lattice.
The parameter e0 is the constant of inverse proportionality
between mussels consumed and the rate of emigration.
Stochastic Processes in the CA Model
In the CA model, mussel recruitment, growth, and mortality are treated as stochastic events. For mussel recruitment and mortality, this was done by choosing an iteration
time step Dt (default value for Dt p 4 days) and assuming
that rates are constant within that time step. This assumption yields an exponential function for the probabilities. The probability of a recruitment event into an
empty cell (x, y) is given by
Pr {Sxy(t ⫹ Dt) p 1FSxy(t) p 0} p 1 ⫺ e⫺jxy(t)Dt.
(11)
Similarly, the probability that a mussel in cell (x, y) dies
is given by
Pr {Sxy(t ⫹ Dt) p 0FSxy(t) 1 0} p 1 ⫺ e⫺mxy(t)Dt.
(12)
Each mussel that dies is assigned a cause of death: background mortality, with probability m 0 /mxy(t), or predatory
mortality, with probability 1 ⫺ m 0 /mxy(t). Mussels consumed by predators in each time step are tracked in variable Ĉ(t), which is used in the computation of predator
emigration rates (see below). If a mussel escapes a random
mortality event, then it may experience a growth event in
that time step. While this assumed ordering of survival
followed by growth seems arbitrary, from a practical consideration, growth occurs slowly on the timescale of our
model iterations, so that this assumption is of little importance. We use a truncated Poisson distribution to
model mussel growth between integer size classes i, which
allows integer increments between time steps that are
greater than 1. The probability that a size-i mussel will
grow to be size i ⫹ j during time increment Dt is defined
as
Pr {Sxy(t ⫹ Dt) p i ⫹ jFSxy(t) p i} p
⫺l xy
e
{
l
j!
1⫺
j
xy
冘
s ⬁,xy⫺i⫺1
kp0
(13)
0 ≤ j ! s ⬁, xy ⫺ i,
e⫺l xy lkxy
k!
0
j p s⬁, xy ⫺ i,
j 1 s⬁, xy ⫺ i,
where l xy p DSxy Dt (from eq. [5]) is the expected growth
increment over the time interval Dt.
Predator immigration and emigration are handled as a
stochastic birth-death process. However, predator movements occur on a much faster timescale than do changes
in mussel size, and these timescales must be handled appropriately in the simulations. For example, the expected
number of predator immigrants is whIDt, which may, in
fact, exceed the theoretical equilibrium number of predators unless Dt is small. On the other hand, if Dt is too
small, then the size density of prey eaten, Ĉ(t), may fluctuate near 0, causing the predator emigration rate to approach infinity and the probability of emigration to approach 1. Therefore, a second, shorter time step of
Dt p Dt/n was used for predator immigration and emigration events in the CA model. During the time interval
Dt, mussel densities remain constant, while predators randomly enter and leave the system every Dt. On this fast
timescale, the probability that j predators enter the system
is given by a Poisson distribution,
Pr {L(t ⫹ Dt) p j} p
e⫺whIDt(whIDt) j
,
j!
(14)
where L(t ⫹ Dt) is the number of predators entering the
system. For emigration, the probability that an individual
predator leaves the system is calculated with equation (10),
618 The American Naturalist
Results
Dynamics of the Mean-Field Model
The solution to the mean-field model gives rise to three
cases: (1) a single stable equilibrium (attracting node) of
high mean mussel size density (upper S ∗; fig. 3A), (2) a
single stable equilibrium (attracting node) of low mean
mussel size density (lower S ∗; fig. 3C), or (3) two stable
equilibria (attracting nodes) of high (upper S ∗) and low
(lower S ∗) mean mussel size density separated by an unstable equilibrium (saddle point; fig. 3B).
At the upper S ∗, mussels are, on average, at a larger,
more predator-resistant size, which corresponds to a low
predator density equilibrium (lower P ∗) because predator
emigration is high when per capita mussel consumption
is low (eq. [10]). The upper S ∗ is the only stable equilibrium (as in fig. 3A) when predator attack rate v is low
(fig. 4A), resistance to predation c is high (fig. 4B), mussel
growth rate b is high (fig. 4D), or predator emigration
Figure 3: Plots of f(S) versus S (eq. [2]) for three values of predator
attack rate v: A, the mean-field model exhibits a single stable upper
equilibrium when v p 0.1; B, the solid line indicates stable upper
(S∗ p 174 mm cell⫺1) and lower (S∗ p 14.5 mm cell⫺1) equilibria
separated by an unstable equilibrium when v p 1 ; the dashed line
indicates transition from a single equilibrium to multiple equilibria;
C, a single stable lower equilibrium for v p 10 . Stable equilibria are
indicated by circles; unstable equilibria are indicated by squares. All
other parameters are at the default values (table 1).
where the summation in the denominator is replaced with
Ĉ(t), the actual summed sizes of mussels consumed by sea
stars during the previous time interval Dt. The probability
that a single predator emigrates during the time interval
Dt equals
ˆ
e p p 1 ⫺ e⫺Dte 0whP(t)/C(t),
(15)
and the probability that j predators emigrate is given by
the binomial distribution
Pr {U(t ⫹ Dt) p j} p
( )
whP(t) j
e P(1 ⫺ e P)whP(t)⫺j,
j
(16)
where U(t ⫹ Dt) is the number of predators leaving the
system and whP(t) is the number of predators at the beginning of the time interval (product of the lattice size
and the predator density). At the end of each time interval
Dt, the new predator density equals
P(t ⫹ Dt) p P(t) ⫹
L(t ⫹ Dt) ⫺ U(t ⫹ Dt)
.
wh
(17)
Equation (17) is iterated n times for every Dt time step.
Figure 4: Bifurcation plots of equilibrium mean mussel size density
(S∗, the roots of eq. [2]), for predator attack rate v (A); predation
resistance c (B); predator immigration I (C); mussel growth rate b
(D); predator emigration constant e0 (E); recruitment rate j (F);
maximum size, s⬁ (G); and background mortality rate m0 (H). Solid
lines represent stable equilibria, and dashed lines represent unstable
equilibria.
Mussel Bed Boundaries as Thresholds 619
rate e0 is high (fig. 4E). At the lower S ∗, mussels are, on
average, at a smaller, more vulnerable size, which corresponds to a high predator density equilibrium (upper
P ∗) because predator emigration is low when per capita
mussel consumption by predators is high (eq. [10]). The
lower S ∗ is the only stable equilibrium (as in fig. 3C) when
predator attack rate v is high (fig. 4A), resistance to predation c is low (fig. 4B), predator immigration I is high
(fig. 4C), mussel growth rate b is low (fig. 4D), or background mortality rate m0 is high (fig. 4H). When the lower
S ∗ is the only equilibrium (fig. 3C), mussels are consumed
almost immediately after recruitment; practically, this is a
no-mussel equilibrium.
For a range of parameter values, both upper and lower
equilibria exist and are separated by an unstable equilibrium (fig. 4), and the system state depends on initial conditions (fig. 5A, 5B). This system is characterized by two
alternative stable states and exhibits hysteresis: a small parameter change (e.g., a decrease in predator attack rate v
from 3 to 1; fig. 4A) can result in a dramatic shift in the
state variable (mean mussel size density moves from the
lower S ∗ to the upper S ∗), but reversing that parameter
change (increasing v from 1 to 3) does not reverse the
effect (mean mussel size density remains at the upper
S ∗). In this model, hysteresis results from size-dependent
predation. Mussel beds with high mean size density are
resistant to predation: predators immigrate into the system
and find few mussels of preferred size to eat, resulting in
a high emigration rate and low predation pressure. Mussel
beds with low mean size density are susceptible to predation: predators immigrate into the system and find many
mussels of preferred size, resulting in a low emigration
rate and high predation pressure that prevents mussels
from reaching more predator-resistant sizes. Both of these
states have positive feedbacks, resulting in hysteresis.
Comparison of the Mean-Field and CA Models
When neighborhood size is equal to the entire arena (i.e.,
a “global” neighborhood) and the landscape is uniform,
the CA model is a stochastic simulator of the ODE model
(fig. 5). For the default parameter values (table 1), the
system displays two stable equilibria separated by an unstable equilibrium (fig. 3B), and the system state depends
on the initial conditions. When initial mean mussel size
density is below the unstable equilibrium, the system converges to the lower equilibrium (lower S ∗; blue and green
lines in fig. 5), which is dominated by small mussels (fig.
Figure 5: Comparing the ordinary differential equation (ODE) model (eq. [1]; A, B) with the cellular-automaton (CA) simulation (C, D)
when the CA model is spatially homogeneous and has a global neighborhood. The upper panels display mean mussel size density through
time in the ODE (A) and CA (C) models; insets display the size-frequency distribution of mussels at the lower (i) and upper (ii) equilibria.
The lower panels display predator density through time in the ODE (B) and CA (D) models. Four initial conditions are plotted: S(0) p
0 (blue), 46 (green), 48 (red), and 200 (black).
620 The American Naturalist
5Ai, 5Ci) and has high predator density (fig. 5B, 5D, upper
P ∗). When initial mean mussel size density is above the
unstable equilibrium, the system converges to the upper
equilibrium (upper S ∗; red and black lines in fig. 5), which
is dominated by large mussels (fig. 5Aii, 5Cii) and has low
predator density (fig. 5B, 5D; lower P ∗). The CA model
duplicates both transient and equilibrium behavior of the
ODEs for mussel density, mussel size distribution, and
predator density.
Effects of Spatially Local Interactions in
Homogeneous Space
Spatially local interactions change the equilibrium dynamics (fig. 6). When predation risk depends on global mean
mussel size density in the CA model, a mussel bed establishing on bare substrate (S(0) p s 0) remains at the lower
equilibrium (fig. 6; dashed black line: r p global), as expected from the ODE model (fig. 5A). However, when
predation risk depends on local mussel size density, a mussel bed establishing on bare substrate (S(0) p s 0) reaches
the upper equilibrium (fig. 6, green line: r p 0; dashed
blue line: r p 1). When r p 0 (fig. 6, green line), predation depends only on individual mussel size: individual
mussels become resistant to predation, and the system
reaches the upper equilibrium (fig. 6i). When r p 1 (fig.
6, dashed blue line), large mussels provide protection to
neighboring mussels and patches of large, predator-resistant mussels form (fig. 6ii) and expand to establish the
upper equilibrium (fig. 6i). When r ≥ 2 (fig. 6, red line:
r p 2, dashed black line: r p global), any size advantage
gained by a mussel is diluted by the smaller size of its
neighbors; as a result, predation risk is high and the system
remains at the lower equilibrium (fig. 6iii). Therefore,
when the environment is spatially homogeneous, very
small neighborhoods (r p 0, 1) allow the system to reach
the upper equilibrium; however, with even moderate-sized
neighborhoods (r ≥ 2), the system behaves as if wellmixed. Note that although per predator predation risk is
local (vxye⫺cSxy), overall predator density is influenced by
all mussels in the arena through predator emigration rate
(eq. [10]).
Boundary Formation along a One-Dimensional
Environmental Gradient
Consider a one-dimensional, linear gradient in predator
attack rate, v p [3, 0], along the Y-axis (fig. 7A), which
represents decreasing predation risk with increasing tidal
height. In the ODE model, the equilibrium solution shifts
from a single stable upper equilibrium in mean mussel
size density to simultaneously stable upper and lower equi-
Figure 6: Neighborhood effects in homogeneous space: cellular-automaton model simulations for neighborhoods r p 0 (green line), 1
(dashed blue line), 2 (red line), and global (dashed black line). Insets illustrate the upper equilibrium (i; upper S∗ , r p 0 at t p 40,000),
the transition from lower to upper equilibrium (ii; r p 1 at t p 6,000 ), and the lower equilibrium (iii; lower S∗ , r p 2 at t p 40,000). The
color bars indicate mussel size S (mm).
Mussel Bed Boundaries as Thresholds 621
Figure 7: Concordance of mussel bed boundary in a cellular-automaton (CA) model and multiple equilibria in an ordinary differential
equation (ODE) model along a gradient in predator attack rate (v). A, One-dimensional landscape gradient in v from a low-intertidal value
of 3 to a high-intertidal value of 0; the color bar indicates the value of v. B, Steady state of a CA mussel bed simulated on the landscape
from A with a linear tidal gradient in v when r p 2 (t p 20 # 103 ); the color bar indicates mussel size (mm). C, Tidal height versus mussel
size density in the CA model; lines show progression through time (every 3 # 103 time steps, thin gray lines) until a stable boundary is
reached (thick black line). D, Stable (solid lines) and unstable (dashed line) equilibrium solutions of the ODE (S∗ ) for each level of v at
the steady state predator density (P p 0.00131 at t p 20 # 103) in B.
libria when solved at values of v p [0, 3] (fig. 4A). When
a CA model with spatially local interactions (r p 2) is
used, a sharp boundary in mussel size emerges on this
linear gradient in attack rate (fig. 7B). Mussels in the high
intertidal experience lower predation risk and grow to
reach maximum size; the boundary forms higher in the
intertidal and shifts downward through time until a stationary boundary location is achieved (fig. 7C). The location of the boundary corresponds to the shift from one
to two stable equilibria in the ODE model (fig. 7D), when
the ODE model is solved with the global predator density
from the CA model arena.
In contrast to dynamics without gradients (fig. 6),
boundary formation in the presence of an environmental
gradient is robust to neighborhood size. For a wide range
of intermediate neighborhood sizes (1 ≤ r ≤ 25; fig. C1,
available online), a boundary forms along the gradient in
predator attack rate that corresponds to the shift from one
to two stable equilibria in the ODE model. The sharpness
and precise location of this boundary depend on neighborhood size (fig. C1).
Effects of Varying Predator Immigration in the
Presence of an Environmental Gradient
Global predator density links the predation risk of all mussels in the arena, while neighborhood effects link per pred-
ator predation risk at smaller scales. The existence of the
mussel bed boundary depends on local interactions, while
the location of the boundary depends on global predator
density. To demonstrate the sensitivity of boundary location to predator density, we varied predator immigration
rate (I), mimicking the removal (lower I) or addition
(higher I) of Pisaster in field experiments (Robles et al.
2009). Halving predator immigration rate (i.e., removing
Pisaster from a mussel bed) halved predator density and
resulted in extension of the mussel bed lower into the
intertidal. Doubling the predator immigration (i.e., adding
Pisaster to a mussel bed) increased predator density by
50% and resulted in retraction of the mussel bed higher
into the intertidal (fig. 8). The resulting Pisaster densities
(0.5, 0.33, and 0.17 m⫺2 for addition, control, and removal,
respectively) and mussel bed dynamics in the model are
concordant with the results of recent field experiments
manipulating Pisaster density and tracking mussel bed response (figs. 1, 3 in Robles et al. 2009).
Emergent Properties in the Spatial Structure
of Mussel Beds
To investigate a field-relevant scenario (Robles et al. 2009,
2010), we examined a two-dimensional arena with the
vertical and horizontal gradients representing the effects
of tidal emergence and wave exposure, respectively, on
622 The American Naturalist
wave exposures ≥ 0.55). At lower wave exposures, only a
single equilibrium exists and the lower edge of the simulated mussel bed is gradual (fig. 9Ai, 9B), corresponding
to a phase shift in a single stable equilibrium (figs. 9C,
D1). The upper mussel bed boundary is gradual for all
wave exposures (fig. 9B), reflecting a phase shift in the
single stable-equilibrium solution as a function of the underlying landscape gradients (figs. 9C, D1).
The lower boundaries of real mussel beds show the same
alongshore trend in location and intensity, but to a lesser
degree. Areas with reduced wave exposure have mussel
bed boundaries that are less distinct and are at higher tidal
heights than modeled boundaries (Robles et al. 2010).
While neighborhood effects remain present in low-waveenergy regions, the maximum benefit afforded by the
neighborhood declines as maximum mussel size declines;
thus, the strength of the neighborhood effects diminishes,
but is not lost, over the gradient.
Figure 8: Mussel bed boundary location for three levels of predator
immigration: i, I p 0.02 (blue line) is a doubling of the immigration
rate that increases predator density by 50%, to P p 0.020 cell⫺1 (∼0.5
m⫺2); ii, I p 0.01 (black line) is the default immigration rate that
results in a predator density of P p 0.013 cell⫺1 (∼0.33 m⫺2); and
iii, I p 0.005 (red line) is a halving of the immigration rate that
decreases predator density by 50%, to P p 0.0067 cell⫺1 (∼0.17 m⫺2).
The main panel plots tidal height (and predator attack rate v) versus
the mean size density of mussels in the cellular-automaton model,
and adjacent panels display the arena at three levels of immigration.
All other parameters are at default values, and r p 2 . All figures are
plotted at t p 20 # 10⫺3 time steps.
recruitment (j), maximum size (s⬁), and predator attack
rate (v), as illustrated in figure 2. Alongshore gradients of
wave energy are ubiquitous in field systems where the swell
impinges on windward rock surfaces and dissipates its
force leeward. Realistic gradients in v and s⬁ can, separately,
result in transitions from single to multiple stable equilibria (fig. 4A, 4G), and both equilibria exist for all meaningful values of j (fig. 4F). Combining gradients in all
three parameters (fig. 2), we solved the ODE model for
the parameter set (jxy, vxy, s⬁, xy) at each location in the
idealized two-dimensional landscape and identified
regions with either a single stable equilibrium or two stable
equilibria (fig. D1, available online). The simulated steady
state is a wedge-shaped mussel bed (fig. 9A) that reflects
patterns observed in the field (this comparison is detailed
in Robles et al. 2010). At high wave exposures, the character and location of the lower mussel bed boundary is
similar to that in the one-dimensional model: the boundary is abrupt and corresponds to the shift from one to
two stable equilibria in the ODE model (fig. 9Aii, 9Aiii,
9B, 9C, D1). This shift from one to two stable equilibria
occurs only at the five higher levels of wave exposure (note
the unstable equilibria indicated by asterisks in fig. 9C for
Discussion
The classic verbal model for the sharp lower boundary of
mussel beds posited a fixed spatial refuge from predators,
suggesting that this dramatic ecological threshold reflects
an equally dramatic underlying shift in predator attack
rate. In contrast, the model presented here posits a dynamic equilibrium, in which this dramatic ecological
threshold results from the switch from one to two stable
equilibria along smooth environmental gradients. This
conceptualization better captures empirical observations
and experiments (Robles et al. 2009), including the crucial
experiment demonstrating the retraction of the mussel bed
with increased Pisaster density (fig. 8), contra expectations
of the fixed-spatial-refuge theory.
The abrupt mussel bed boundary in the CA model corresponds to the transition from one to multiple equilibria
in the ODE model (figs. 7C, 7D, D1) when the ODE is
solved at each level of predator attack rate using the global
predator density from the CA model. The alignment of
the CA boundary and the ODE equilibria transition is
noteworthy because the ODE predictions are mean-field
solutions of mussel size density assuming constant predator density, while the CA model is a dynamic, two-scale
system with global feedback through predator density and
local feedback through mussel size density.
An important aspect of boundary formation in the onedimensional CA model is the gradual downward expansion
of the mussel bed through time (fig. 7C): starting in an
empty arena, mussels grow to large size in low-predationrisk areas and then confer decreased predation risk to
neighboring mussels, allowing expansion into riskier areas.
In the CA model, this decreased risk arises in two ways:
(1) locally, new recruits at the edge of the mussel bed have
Mussel Bed Boundaries as Thresholds 623
Figure 9: Mussel bed formation in a cellular-automaton (CA) model simulation and an ordinary differential equation (ODE) model varying
in tidal height (Y-axis) and wave exposure (X-axis). A, Established mussel bed in the CA model with r p 2 ; the color bar indicates individual
mussel size (mm). B, Mean mussel size density in the CA model at 10 levels of wave exposure, scaled from 0 to 1 (see key). C, ODE
equilibrium solutions at parameter values corresponding to the same tidal heights and wave exposures as in B (see key in B), solved at the
steady state predator density in A; unstable equilibria are plotted as black asterisks. The insets in A magnify the lower boundary at x p
0.35, (i), x p 0.65 (ii), and x p 0.95 (iii), indicated by white arrows.
decreased predation risk because of protection from large
neighboring mussels, and (2) globally, as more of the arena
is occupied by larger, less-preferred mussels, predator consumption declines and predator emigration rates increase,
leading to lower global predator densities. As the ODE
model is solved with declining predator densities from the
CA model, the boundary between single and multiple
equilibria shifts downward to higher predator attack rates
(fig. 7D). This mussel bed expansion has dynamics similar
to those of an invasion front with Allee effects (Keitt et
al. 2001): the invasion expands when the benefits accrued
from high-density neighboring cells within the front overcome the costs of low density in cells adjacent to the front,
and the front stabilizes when the benefits accrued from
high-density neighbors no longer outweigh the costs of
the low-density neighbors (Keitt et al. 2001). This transition from expansion to stability is equivalent to the transition from single to multiple equilibria (dashed line to
black line in fig. 3B; Keitt et al. 2001). In our model, the
mussel bed boundary ultimately stabilizes where the
buildup of large mussels on the boundary no longer outweighs the increased risk in adjacent cells.
The tendency for local interactions to change dynamics
from the mean-field predictions depends on the landscape.
On a homogeneous landscape (fig. 6), equilibrium be-
havior is highly sensitive to the scale of neighborhood
effects: only the smallest neighborhoods (r p 0, 1) deviate
from well-mixed behavior by converging on the upper
equilibrium. On more realistic heterogeneous landscapes
(here, a one- or two-dimensional gradient), boundary formation is robust to a wide range of neighborhood sizes
(fig. C1), even though the well-mixed model remains at
the lower equilibrium (fig. C1A; r p global). The difference between the ODE predictions based on the CA model
predator density (fig. 7D) and the full equilibrium solution
of equation (3) (fig. 4A) demonstrates the importance of
local interactions. In both the homogeneous and the heterogeneous landscapes, local interactions allow mussels to
occupy a larger proportion of the entire arena. The scale
of local interactions also influences the character of the
lower boundary (fig. C1C–C1G). However, subtle changes
in boundary character with neighborhood size may be less
important than larger changes over the boundary range
(see fig. 9 and discussion below) and less important in the
field because of small-scale heterogeneities in the physical
features in the intertidal. More important is that boundary
formation is robust over a wide range of neighborhood
sizes. Future work should consider the effects of smallscale landscape heterogeneities on mussel bed structure
624 The American Naturalist
and boundary formation when those heterogeneities are
at a scale similar to that of local interactions.
The full two-dimensional simulation (fig. 9) demonstrates several patterns that are concordant with field observations, including boundary location and fragmentation (Robles et al. 2009, 2010). In this dynamical system,
we find both abrupt and continuous boundaries, where
the abrupt boundaries correspond to bifurcations from
one to multiple equilibria (figs. 7, 9C, D1) and the continuous boundaries correspond to phase shifts in a single
equilibrium. Previous studies on species range limits (Holt
and Keitt 2000; Keitt et al. 2001; Holt et al. 2005) have
noted how Allee effects can result in sharp species boundaries, in contrast to more gradual species boundaries that
result from changes in colonization and extinction rates.
This model posits this same variation in the nature of
species boundaries over a much smaller spatial scale. We
conceived this model at the scale of an intertidal mussel
bed (tens of meters); in this system, the neighborhood
effects (decreased predation with increased mussel size)
vary in magnitude over a few meters and result in distinct
boundary types within the same dynamical system. In areas
of the landscape with large maximum mussel size and
moderate predator attack rate (lower right in the twodimensional gradients; fig. 2), neighborhood effects result
in multiple equilibria (figs. 9C, D1B) and therefore sharp
boundaries (fig. 9A). In other parts of the landscape, the
same underlying model sharing the same predator population results in a single equilibrium (figs. 9C, D1A),
which has continuous boundaries corresponding to the
slow change of underlying parameter values (fig. 9A). This
model provides an example, with corresponding empirical
data (Robles et al. 2009, 2010), in which the same underlying model gives rise to species boundaries of different
characteristics because of fundamentally different threshold dynamics in different regions of the landscape.
Like other models that generate spatial ecological
thresholds from bifurcating equilibria (e.g., Wilson and
Nisbet 1997; van de Koppel et al. 2005; Zeng and Malanson
2006), this model relies on locally positive and globally
negative feedback. Wilson and Nisbet (1997) considered
this problem directly in a general cell-occupancy model
of intraspecific cooperation and competition and demonstrated that positive local interactions can generate a
spatial threshold on a continuous gradient. The model
presented here advances their work by connecting the theory to an explicit empirical system; in doing so, we demonstrate that boundary formation is robust to the additional complications of size-structured predator-prey
interactions over a range of neighborhood sizes. More generally, the model presented here is an example of “classical
criticality” as defined by Pascual and Guichard (2005), in
which disturbance (predation on mussels) is well mixed
and disturbance intensity depends on the local density of
susceptible individuals. A characteristic of classical criticality is the presence of a threshold along a gradient of
disturbance, as demonstrated here by the abrupt boundary
between upper and lower mussel bed equilibria.
Combined with previous papers (Robles and Desharnais
2002; Robles et al. 2009, 2010), this work indicates a concordance between a dynamic predator-prey model and
field experiments and observations, providing a new paradigm for understanding mussel bed boundaries. This
concordance of field observation and model predictions
at small spatial scales is based on empirical patterns of
covariance between mussel growth rates, predator attack
rates, and settlement patterns within mussel beds, as represented in figure 2. On regional spatial scales, however,
these parameters may covary in different ways (Menge et
al. 1994, 2004). Future work should focus on expanding
predictions of this model to regional-scale patterns of
boundary formation and location in the intertidal.
Acknowledgments
This work was improved by discussions with C. Garza, F.
Guichard, R. Nisbet, and J. van de Koppel. The work was
supported by National Science Foundation grants HRD0317772, OCE 0089842, and EF-0827595.
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Associate Editor: Marc Mangel
Editor: Ruth G. Shaw